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MatPlus.Net Forum General A puzzle puzzle

### A puzzle puzzle

https://chess.stackexchange.com/questions/33569/a-new-years-math-riddle

You probably all know the "How often can a position be repeated" puzzle,
but it critically depends on the side having the move be able to ep.
And constructing a position with an ep right AND both KRR in the initial
position seems hopeless to me...but I'm no retro expert. Is that possible?
Or at least an SPG to weasel out? That should be easy: 1.d4 e5 2.Lh6 e4 3.f4

-

The first diagram on Tim's page is no longer a record for the maximum number of positions possible. Instead, that record (including the board rotation trick) is held by GM Jonathan Mestel, see https://pdb.dieschwalbe.de/P1359129. The capture-counting is quite subtle: I hope my solution of his problem is accurate! He picked the one legal configuration in which the number of pieces captured per side is determined, whichever way round the board is. Tim's diagram is legal, even with the queens present: there are were 4+4 captures, whichever way round is the board.

I guess an SPG where the position offers all castling rights + e.p. can be achieved in 2.0 but it needs a capture e.g. 1.a4 b5 2. axb5 a5. Here bBc8 is released to perform for parity-switching services.

Ah, but I have to correct: The question is not "How many positions
are in one diagram" but "How often can the diagram be repeated
without 3-fold rule". Obviously it that setting it is not relevant
WHICH ep is possible - it is only one pawn that has the right
in any of the positions.

Hi Hauke yes I’m aware of this sorry if I wasn’t clear. Maybe I should have put my two comments in a different order. First my tiny remark about Joost’s SPG (pertaining to the 22 problem), and secondly the remark about Mestel's related problem improving on Tim Krabbé intro to the 22.

EDIT: Inspired by Rewan, I've posted another question: https://chess.stackexchange.com/questions/33707/an-expanded-new-years-math-riddle

The Dead Reckoning rule destroys many easy attempts of making an unique SPG.
I remember puzzles that forced both players to cooperate to reach the same position again, but that position was given. Your question is different, as pawns can move and the position after the pawn move can be repeated, i.e. it doesn't have to be a specific but any position to be repeated.

My favourite (and judging by the comments, also Andrew's favourite) 50 move/3-fold repetition composition: https://pdb.dieschwalbe.de/P1011799

Thanks Siegfried,

QUOTE
The Dead Reckoning rule destroys many easy attempts of making an unique SPG

Eh? Please give me a precise example what you're talking about. DP rule is very rarely involved in proof games. There are a few massacre SPGs but that's about it. It does have utility, but only in very narrow circumstances.

And DP rule is Roger Irrelevant for this problem. Firstly because at any point a player can move a pawn, zeroing 3/5-rep and draw by 50/75 moves. But mainly because it's undefined whether DP rule can see the conventions. The content justifies it in Caillaud's problem.
Maybe in the future a clear paper will be written which explains how all these can come together, but until that point, the composer can choose, I'd let DP see the conventions only if the particular content demands it. Post Article 17A nerfing, DP may not "destroy" problems but it certainly challenges the phrasing of a number of the conventions, so why bring in boring semantics?

QUOTE
I remember puzzles that forced both players to cooperate to reach the same position again, but that position was given. Your question is different, as pawns can move and the position after the pawn move can be repeated, i.e. it doesn't have to be a specific but any position to be repeated.

I mean that you should find a position yourself and revisit it lots of times. But there are thousands of very obvious positions which would work, for example the Tim Krabbe one. That's not the challenge for most of the questions, which is why I don't give a particular position right at the start. It's only in Q4 where I ask can you find such a starting position which is also an SPG for the 5-fold case, just as Hauke did above for the 3-fold case.

Hauke's SPG for 3-fold *doesn't* work for the 5-fold case. And there's another trick too.

Andrew,

Q4:
(= 4+3 )

Shortest way to reach a 5-fold repetition is 1. Ra8 Kb7 2. Ra6 Kb8 etc, so that's a unique SPG for the 5-fold repetition.

@Joost this isn't an SPG at all, it's a help-draw.

Q4 mentions an SPG which will start from the game array as usual, and move to a position where all castling rights remain, en passant, etc. Like the one Hauke did for 3-rep, but that one won't quite work here.

I can't remember ever having done anything with the 3-fold repetition and SPGs. Are you referring to Michel's composition?

And I remembered this one:
Andrei Frolkin, Sergei Tkachenko
Die Schwalbe 1997
(= 13+13 )

SPG 23.0 in which black played 19... b4. Result?

For the 5-fold repetition this should be 'SPG 27.0 in which black played 19...b4'.

@Joost sorry I mean Hauke: 1.d4 e5 2.Lh6 e4 3.f4. I think your post 2 in this thread had an SPG for a short time maybe, but it's empty now. I'll fix my mentions.

Yes I'm familiar with the 23.0 PG from years ago. It's rare that there is fundamentally different behaviour between 3-fold & 5-fold repetition, which is one reason the chess.stackexchange problems are interesting, I hope

(= 4+1 )

s"="4

Michel Caillaud did a s"="2 in Strategems years ago where you could prove the previous 2 half-moves, and the next two half-moves were also forced, leading to a 3-fold repetition, but I can't find it in Winchloe, PDB or yacpdb.

By the way, if you're really nitpicky, your puzzle doesn't work, since you're asking for composed games, so automatically the composition rules apply, which end the game immediately after the 3rd repetition or when the 50-move rule triggers.

@Andrew:
Oops, I thought you meant from a specific position.
But anyway, forcing a threefold or fivefold repetition from some the beginning in an unique way is difficult.

(16) Posted by Andrew Buchanan [Sunday, Jan 10, 2021 07:17]

Somehow there is still some misunderstanding. Let me restate.

Original Q0: what is the max number of occurrences of a single diagram possible under 3-Rep rule? Answer: 22
New Q1: ditto under 5-Rep rule?
New Q2: ditto under 3-Rep & 50M rules?
New Q3: ditto under 5-Rep & 75M rules?
Hauke Q0a: can you find a diagram satisfying Q0, which has a unique proof game to the *first* occurrence?
New Q4: Same as Hauke's question, but for Q3 instead of Q0.

Assume:
- Players co-operate to maximize the number of occurrences.
- For Q0, Q2, Q0a, claims for 3-Rep & 50M are mandatory, otherwise they are optional.
- No involvement of conventions.